## Positioning Students as Mathematically Competent

Promoting and valuing students’ participation in mathematical discourse—sharing their reasoning; creating, critiquing, and revising arguments; and engaging in collaborations aimed at making sense of and using mathematical ideas—is a way of positioning them as being mathematically competent. In order to ensure that each and every student not only understands and can make use of foundational mathematics concepts and relationships but also comes to experience the joy, wonder, and beauty of mathematics, we must position each and every student as mathematically competent. This requires creating classroom structures—norms and routines—that support students to take risks to engage in discourse and to see themselves as capable and worthy of being heard. In doing so, students’ mathematical identities are connected to their participation in a set of productive practices and processes of doing mathematics. Aguirre, Mayfield-Ingram, and Martin (2013) define mathematical identity as “the dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives” (p. 14).

However, in too many mathematics classrooms mathematical competence is assigned solely on the basis of quickness and correctness, giving the mistaken impression that only some students are “good at math.” This creates an environment where students’ mathematical reasoning goes unexamined and unvalued; consequently, little is known about how they make sense of mathematics, how they use their mathematical understanding in developing solutions, and why their solutions do or do not make sense. Correct answers matter but not as indicators of who is able to do mathematics. Engaging in mathematical discourse is essential for developing mathematical identity and should be recognized as a better indicator of mathematical competence. In what ways must our classrooms and lessons change to promote positive mathematical identities for each and every student?

To get an understanding of positioning students as competent, I invite you to watch Video Clip One from the Bike and Truck Task found in NCTM’s *Principles to Actions* Toolkit. The video clip is drawn from a high school algebra 1 class but practices modeled in this lesson have strong relevance across all grade bands. To set up the discussion the teacher, Ms. Shackelford, invented a fictional student, Chris, to help her students focus and clarify their thinking about the graphical representation of the position of the truck as a function of time. After watching the video, think about the following questions:

- What norms and routines must have been established and practiced to allow students to engage in the sort of mathematical discourse that positioned each of them as mathematically competent?
- How do the forms of participation move the students forward in their thinking about the mathematics?
- What would happen to the students’ respective mathematical identities if this same task unfolded in a different classroom in which Jacobi is told immediately his reasoning is incorrect and Charles is told he is correct?

The video clip is an illustration of how Ms. Shackelford engaged students in reasoning and sense making through a routine of listening to and critiquing others’ reasoning. Ms. Shackelford positioned Jacobi (yellow shirt) and Charles (maroon shirt) as capable contributors to mathematical discussion. Jacobi’s reasoning did not fit the graphical representation of the truck but he was highly participatory and was able to interact with Charles, whose reasoning did fit the graphical representation of the truck. In the clip we see Jacobi and Charles engaged in public sense-making by sharing their mathematical thinking with their peers and Ms. Shackelford. By publicly making the interaction between Jacobi and Charles worthwhile, Ms. Shackelford positions both students as having mathematical competence through their participation. Their ideas were welcomed and used to build mathematical understanding. When students share and value their mathematical ideas through processes of mathematical discourse, they move away from mathematics competence as producing correct answers quickly and toward mathematics competence as participatory.

Ms. Shackelford conducted this lesson in April of the academic year, and it appeared that the social norm in her classroom had been firmly established and that her students were well aware of, and comfortable with, her expectations that they would explain their thinking, respectfully critique others’ reasoning, and make mathematical connections. The lesson in Ms. Shackelford’s classroom also modeled intellectual authority as being shared between the teacher and students. As students author ideas, decide and justify whether particular ideas are reasonable, and press one another for explanations, they take on forms of intellectual authority that support collaborative mathematics teaching and learning (Langer-Osuna, 2017).

Positioning students as mathematically competent must happen with clarity and consistency to have a long-lasting positive impact on their mathematical identities (Munson, 2018). The questions below are a start for reflecting on how students might be positioned as mathematically competent in your classroom.

- How do I create classroom norms and routines that support students to take risks to engage in mathematical discourse?
- In what ways are students’ mathematical ideas shared and valued?
- How do my teaching practices communicate to each and every student that their ideas matter?
- In what ways is intellectual authority distributed in my classroom?
- How do my teaching practices use students’ ideas to guide them to important mathematical insights and understandings?

I encourage you to use the questions to reflect on your classroom and teaching practices. Please share your successes and challenges on MyNCTM.org.

Robert Q. Berry, III

NCTM President

**References**

Aguirre, Julia, Karen Mayfield-Ingram, and Danny Martin. *The impact of identity in K-8 mathematics: Rethinking equity-based practices*. Reston, VA: National Council of Teachers of Mathematics, 2013.

Langer-Osuna, Jennifer M. "Authority, identity, and collaborative mathematics." *Journal for Research in Mathematics Education* 48.3 (2017): 237-247.

Munson, Jen. "Two Instructional Moves to Promote Student Competence." *Teaching Children Mathematics* 24.4 (2018): 244-249

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## All Comments

Shelly Jones- 1/8/2019 3:19:06 PMI will use these questions in my talk with educators at the AMS/MAA Joint Meetings in Baltimore next week.

This topic is so important and one that I've been reflecting on alot.

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Kerry Faber- 8/8/2018 12:19:32 PMThank you for the ideas you have put forwad in this article. I have been encouraging teachers to deepen their understanding of the strands of mathematical proficiencies. While the beginning of your article made me think of the importance of productive disposition I appreciate how you wove strategic competence and adaptive reasoning throughout.

I especially like the wording of your reflective questions on how we can help position our students to be mathematically competent. I look forward to sharing these for discussion and planning purposes.

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Kimberly Yoak- 8/7/2018 12:18:28 PMThank you for sharing this vitally important perspective in your column, and thank you for citing several different authors who are also writing about the same ideas. It is really important for educators and all of us to see that these practices apply (and are effective) across grade levels and across zip codes.

I feel that my work in mathematics education, when based on the framework shared in your column and other related literature, is one way of fostering equity in our society well beyond the mathematics classroom. When a student feels valued and appreciated and sees that every other student is valued and appreciated, and when this happens in every classroom, that student leaves school with far more value, appreciation, and empathy for those in the world around (repetition in this sentence intended!). Thank you for being one who is voicing this call to action on a national level!

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John Wenger- 8/1/2018 4:00:24 AMThis post is very interesting, and it raises a huge number of points to be considered. I thank you for thinking these things through, and you have given me a lot to think consider. I would like to say that the video of Ms. Shackelford shows her attempting to use the Socratic method in teaching, which is a wonderful way to teach if the teacher has the time and ability to do so. When I taught in a college setting, we had to go at least twice as fast as a high school class because we had to cover in an 18-week semester what students in high school have an academic year of 40 weeks or so to learn. I like the idea of having more time to learn, enabling students to learn at their own pace rather than at an arbitrary pace, but it is impossible given the restrictions we had. (I tried to figure out ways to do this, but I simply couldn't get any support.)

You argue that promoting and valuing each student's participation in mathematical discourse is a "way of positioning them as being mathematically proficient." I would say that it is a way of

makingthem mathematically proficient, which is not the same thing, nor is it always possible. I always started my classes with the notion that everyone was going to understand the material, even though I knew it wouldn't happen (except in my first calculus class back in 1969, but I only had two students in it). I agree that we should try to make every student think they are worthy of being heard, but it isn't an easy thing to do, and in a college structure it is generally impossible to let everyone speak on every subject if one is required to cover all the material.Here is a key sentence you wrote: "However, in too many mathematics classrooms mathematical competence is assigned solely on the basis of quickness and correctness, giving the mistaken impression that only some students are 'good at math.'” I agree that quickness should not be valued, which is why I always gave my students as much time as they wanted on exams. This is also a way of counteracting the fear so many students have of this subject; if they freeze up, the extra time can be used to let them relax and look at the problems with less panic. Unfortunately, in many cases it didn't work, although at least students didn't feel that they had been treated unfairly when their grades were less than desired by them or me.

Correctness is another matter. It is fine to value the views of all students, and it seems that there are some mistakes that everyone must make before they learn (the sum of the squares is the square of the sum, for example, or the ever-present student rule of universal cancellation regardless of what operations are being used on the top and bottom of a fraction). On the other hand, there is a time of reckoning, and at that time correctness is king, although I do believe in looking at a student's work; this way, if they do everything right except saying two times three equals five somewhere, I let such mistakes go or grade very little off if I realize that the mistake was silly. In short, I look at the whole of what a student does to see if he has the correct understanding. All of this is a matter of judgment, but that is why we are credentialed in our knowledge of the subject, allowing us to make such decisions in a reasonable fashion. I think this is what you mean when you say, "Engaging in mathematical discourse is essential for developing mathematical identity and should be recognized as a better indicator of mathematical competence." I agree with this, both in imparting knowledge and in evaluating it, although when it comes to imparting the knowledge, as I said previously, we are often severely limited in our ability to do this due to time contstraints.

I'm not sure how much this has to do with tracking, however. The fact is, there are some students who are simply not fit for mathematics no matter how good their backgtound, no matter how much time is lavished on them, and so forth. Putting students into fairly homogenized groups with respect to a variety of criteria makes sense both for the better students and the ones not as good. People have different talents, and most people have some things they do well and some things they do not, even in the same discipline. For example, when it comes to three-dimensional calculus, I have real problems seeing the material at times, and I have had students help me graph some curves because of my inability to visualize and reproduce three-dimensional objects. I have gotten better at it with time, but it just isn't one of my talents. There are some students who do better in applied mathematics, whereas my strength is in pure mathematics, making the teaching of applications a challenge for me. All of us have strengths and weakmesses. even in our own fields. As to other fields, some people have no talent in mathematics no matter how much one values their participation, while for others they succeed without much effort. To ignore these natural differences, to expect that they can be overcome with more effort, is dangerous in my opinion, because it creates expectations that cannot be met. We should try our best to overcome these problems, but we should recognize reality at the same time.

John C. Wenger

Professor Emeritus

Harold Washington College (One of the City Colleges of Chicago)

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John Plencner- 7/30/2018 2:58:55 PMAfter teaching high school for fifteen years, it seems to me that we are disconnecting with students (mathematically) somewhere between 5th and 9th grades. We really need to have a discussion on why so many students that are successful and enjoy math the first three or four years of school arrive in high school disillusioned and disliking math and convinced that they cannot be successful.

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Janet Herrelko- 7/29/2018 6:23:30 AMIf teachers were given time to collaborate to create integrated lessons, detracking and positioning would be the fundamental organization of the students. Problem-based learning could be created by the teachers to have studets solve real world problems pertinent to them. These plans would reflect the Mathematical Practices of the Common Core, the Teaching Practices of Principles to Actions. Love the integration of all these President's Messages! Thanks so much for spreading the news!

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Brian Lawler- 7/27/2018 9:52:12 PMConsider providing access to the Bike & Truck Task video to non-members?

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Kimberly Johnson- 7/27/2018 2:09:37 PMI appreciate receiving this just prior to the start of a new school year. I am no longer working with K-12 teachers but I plan to use this with my preservice teachers. How timely it will be, and thought provoking, as they begin their field experience with upper elementary students. Although this is a high school algebra classroom, it is so very relevant to any K-12 classroom. Thanks for the inspiration and hope!

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Cynthia Slaton- 7/26/2018 9:30:10 PMThis is relevant. An overlooked skill.

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Sue Leedes- 7/26/2018 11:29:35 AMEvery teacher of mathematics should read and believe this.

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DeAnn Huinker- 7/25/2018 7:03:56 PMHere is the link to the Priinciples to Actions Toolkit: https://www.nctm.org/ptatoolkit/

Then scroll to the High School section, Bike and Truck Task.

For the specific video clip referenced: https://www.nctm.org/Conferences-and-Professional-Development/Principles-to-Actions-Toolkit/The-Case-of-Shalunda-Shackelford-and-the-Bike-and-Truck-Task/

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Robert Berry- 7/26/2018 12:21:14 AM## Reply processing please wait...

Robert Berry- 7/26/2018 12:22:32 AMDeAnn, Thanks for making sure the links were correct.

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Karen Fuson- 7/25/2018 5:15:22 PMAn excellent column!!! This is such an important topic. And as each teacher establishes these norms and students experience such participation and value, they carry the norms into their next classroom.

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Robert Berry- 7/26/2018 12:20:53 AMDr. Fuson, Thank you so much. I have be a long admirer of your work.

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Kim Meyer- 7/25/2018 5:12:10 PMHow about including links? Teachers are busy and having to look for the link, probably means 75% wll not go any further.

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Robert Berry- 7/26/2018 12:17:48 AMThank you for the feedback. The links to the video should be working now and the links to the two articles in the reference should work as well.

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